# Properties

 Label 400.6.a.o Level $400$ Weight $6$ Character orbit 400.a Self dual yes Analytic conductor $64.154$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 400.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.1535279252$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{241})$$ Defining polynomial: $$x^{2} - x - 60$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 25) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{241}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -10 - \beta ) q^{3} + ( -100 - 2 \beta ) q^{7} + ( 98 + 20 \beta ) q^{9} +O(q^{10})$$ $$q + ( -10 - \beta ) q^{3} + ( -100 - 2 \beta ) q^{7} + ( 98 + 20 \beta ) q^{9} + ( 98 - 25 \beta ) q^{11} + ( -180 + 16 \beta ) q^{13} + ( 745 - 68 \beta ) q^{17} + ( 1590 + 35 \beta ) q^{19} + ( 1482 + 120 \beta ) q^{21} + ( -780 - 6 \beta ) q^{23} + ( -3370 - 55 \beta ) q^{27} + ( -1960 - 40 \beta ) q^{29} + ( 548 + 550 \beta ) q^{31} + ( 5045 + 152 \beta ) q^{33} + ( 1010 + 192 \beta ) q^{37} + ( -2056 + 20 \beta ) q^{39} + ( 13877 + 200 \beta ) q^{41} + ( 1500 + 1064 \beta ) q^{43} + ( -12880 - 772 \beta ) q^{47} + ( -5843 + 400 \beta ) q^{49} + ( 8938 - 65 \beta ) q^{51} + ( -13490 + 376 \beta ) q^{53} + ( -24335 - 1940 \beta ) q^{57} + ( -5980 + 980 \beta ) q^{59} + ( -12198 + 1000 \beta ) q^{61} + ( -19440 - 2196 \beta ) q^{63} + ( 20030 + 793 \beta ) q^{67} + ( 9246 + 840 \beta ) q^{69} + ( 43648 - 500 \beta ) q^{71} + ( -35145 + 556 \beta ) q^{73} + ( 2250 + 2304 \beta ) q^{77} + ( -32740 - 2510 \beta ) q^{79} + ( 23141 - 940 \beta ) q^{81} + ( -46290 + 429 \beta ) q^{83} + ( 29240 + 2360 \beta ) q^{87} + ( -36405 - 5220 \beta ) q^{89} + ( 10288 - 1240 \beta ) q^{91} + ( -138030 - 6048 \beta ) q^{93} + ( -63070 + 5472 \beta ) q^{97} + ( -110896 - 490 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 20q^{3} - 200q^{7} + 196q^{9} + O(q^{10})$$ $$2q - 20q^{3} - 200q^{7} + 196q^{9} + 196q^{11} - 360q^{13} + 1490q^{17} + 3180q^{19} + 2964q^{21} - 1560q^{23} - 6740q^{27} - 3920q^{29} + 1096q^{31} + 10090q^{33} + 2020q^{37} - 4112q^{39} + 27754q^{41} + 3000q^{43} - 25760q^{47} - 11686q^{49} + 17876q^{51} - 26980q^{53} - 48670q^{57} - 11960q^{59} - 24396q^{61} - 38880q^{63} + 40060q^{67} + 18492q^{69} + 87296q^{71} - 70290q^{73} + 4500q^{77} - 65480q^{79} + 46282q^{81} - 92580q^{83} + 58480q^{87} - 72810q^{89} + 20576q^{91} - 276060q^{93} - 126140q^{97} - 221792q^{99} + O(q^{100})$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 8.26209 −7.26209
0 −25.5242 0 0 0 −131.048 0 408.483 0
1.2 0 5.52417 0 0 0 −68.9517 0 −212.483 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.6.a.o 2
4.b odd 2 1 25.6.a.d yes 2
5.b even 2 1 400.6.a.w 2
5.c odd 4 2 400.6.c.n 4
12.b even 2 1 225.6.a.l 2
20.d odd 2 1 25.6.a.b 2
20.e even 4 2 25.6.b.b 4
60.h even 2 1 225.6.a.s 2
60.l odd 4 2 225.6.b.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.6.a.b 2 20.d odd 2 1
25.6.a.d yes 2 4.b odd 2 1
25.6.b.b 4 20.e even 4 2
225.6.a.l 2 12.b even 2 1
225.6.a.s 2 60.h even 2 1
225.6.b.i 4 60.l odd 4 2
400.6.a.o 2 1.a even 1 1 trivial
400.6.a.w 2 5.b even 2 1
400.6.c.n 4 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 20 T_{3} - 141$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(400))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-141 + 20 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$9036 + 200 T + T^{2}$$
$11$ $$-141021 - 196 T + T^{2}$$
$13$ $$-29296 + 360 T + T^{2}$$
$17$ $$-559359 - 1490 T + T^{2}$$
$19$ $$2232875 - 3180 T + T^{2}$$
$23$ $$599724 + 1560 T + T^{2}$$
$29$ $$3456000 + 3920 T + T^{2}$$
$31$ $$-72602196 - 1096 T + T^{2}$$
$37$ $$-7864124 - 2020 T + T^{2}$$
$41$ $$182931129 - 27754 T + T^{2}$$
$43$ $$-270585136 - 3000 T + T^{2}$$
$47$ $$22262256 + 25760 T + T^{2}$$
$53$ $$147908484 + 26980 T + T^{2}$$
$59$ $$-195696000 + 11960 T + T^{2}$$
$61$ $$-92208796 + 24396 T + T^{2}$$
$67$ $$249648291 - 40060 T + T^{2}$$
$71$ $$1844897904 - 87296 T + T^{2}$$
$73$ $$1160669249 + 70290 T + T^{2}$$
$79$ $$-446416500 + 65480 T + T^{2}$$
$83$ $$2098410219 + 92580 T + T^{2}$$
$89$ $$-5241540375 + 72810 T + T^{2}$$
$97$ $$-3238386044 + 126140 T + T^{2}$$
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